Method for control and coordination of independent tasks using benders decomposition

ABSTRACT

A method for supervisor control and coordination of independent tasks to be performed by a plurality of independent entities in which a plurality of schedules for performance of the independent tasks is generated and submitted to a master coordinator for approval and/or disapproval, resulting in generation of a plurality of approval and/or disapproval decisions by the master coordinator. The approval and disapproval decisions are returned to the independent entities which, in turn, adjust the schedules for which a disapproval decision has been returned, resulting in an adjusted schedule. The adjusted schedule is then returned to the master coordinator for reconsideration. These steps are repeated until all of the schedules have been approved.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] This invention relates to the use of Benders decomposition, atechnique for solving multi-stage stochastic linear programming problemsby decomposing them into a set of smaller linear programming problems,for supervisory control and coordination of independent tasks to beperformed by a plurality of independent entities. More particularly,this invention relates to the application of Benders decomposition tothe problem of scheduling maintenance in deregulated power systems.

[0003] 2. Description of Prior Art

[0004] The electric power industry is moving toward a deregulatedmarket-based operation in which the unbundling of services willestablish independent entities such as power generators (GENCOs),transmission providers (TRANSCOs), independent customers (DISCOs) andindependent system operators (ISOs). In the past, the first threeentities were monopolized, owned and operated by regulated electricpower companies. However, in the market-based structure, the supervisorycontrol and coordination of these independent entities will be a vitaltask for power systems operators.

[0005] In response to the electric industry restructuring, newphenomena, new circumstances, new risks, and new tools have emerged.Some of them arise due to the lack of experience with newborn issues,while others come as a necessity for the new structures. The electricpower market differs from other commodity markets in that electric powercannot be stored and must be consumed as generated. This conditiondrastically changes the nature of supervisory control and coordinationin this industry as compared with that in other industries. As a result,applying theorems and models of other commodities to electricity havefrequently misled participants of restructured electricity markets. Inthis regard, a systematic approach which will provide the propermechanism as individual companies schedule their own activities, satisfytheir own constraints and optimize their own objectives is desirable.From the customers perspective, maximizing the system availability andreliability and minimizing the cost of power delivery are desirableobjectives which require coordination and control among these companies.

[0006] Supervisory control may encompass different applicationsincluding the coordination of short-term and long-term maintenancescheduling of facilities in several independent power companies, fueldelivery and scheduling, and provision of ancillary services (dispersionof reserve capacity among power companies to sustain the continuity ofservice in the case of a contingency). It will be apparent thatachievement of these objectives requires addressing various conflictingobjectives, such as cost versus reliability, in power companies. It isimperative to maintain a certain level of reliability in a power systemwhile individual companies within the system try to minimize their owncost of operation and maximize their revenues. Thus, it is clear that asystematic coordination and supervisory control is essential to assurethe reliability of power delivery and to minimize power outagesthroughout the network.

[0007] Numerous methods relating to the planning and scheduling ofmaintenance for a variety of systems including power generation anddistribution systems are disclosed by the prior art. For example, U.S.Pat. No. 5,798,939 to Ochoa et al. teaches a computer workstation-basedinteractive tool for assessing the reliability of power systems, whichtool can be used to determine the effect on the reliability of bothsubstations and bulk generation and transmission systems of systemadditions, design alternatives, maintenance practices, substationconfigurations, and spare part policies. U.S. Pat. No. 5,970,437 toGorman et al. teaches a program for computerized management of plantmaintenance which provides graphic representations of the mechanical andelectrical systems of an operating plant where all of the components aretreated as “objects” and associated through a relational database withtext files providing component specifications, product identifications,component operating status, and all directly interconnected componentsincluding data as to flow direction and relative position in the system.For maintenance and service, each component is associated with existingand future specified work orders, and service requirements includingconsideration of probable life expectancy and the like. See also U.S.Pat. No. 5,311,562 to Palusamy et al. which teaches an integrated plantmonitoring and diagnostic system for shared use by the operations,maintenance and engineering departments of a nuclear power plant; U.S.Pat. No. 4,843,575 to Crane which teaches an interactive, dynamic,real-time management system comprising a plurality of powered systemsand a central management facility; and U.S. Pat. No. 6,006,171 to Vineset al. which teaches a computerized maintenance management system forthe process control environment which integrates a computerizedmaintenance management system with a process control system.

SUMMARY OF THE INVENTION

[0008] It is one object of this invention to provide a method forsupervisory control and coordination of independent tasks in thederegulated electric power industry aimed at minimizing the possibilityof blackouts, minimizing the cost of power delivery to customers,improving the system availability, and responding to environmental andregulatory concerns as individual power companies try to maximize theirrevenues.

[0009] It is another object of this invention to provide a method forsupervisory control and coordination of independent tasks in thederegulated electric power industry which enables individual companiesto schedule their own activities, satisfy their own constraints andoptimize their own objectives while providing necessary coordination andcontrol among these companies to maximize the system availability andreliability, and minimize the cost of power delivered to customers.

[0010] These and other objects of this invention are addressed by amethod for supervisory control and coordination of independent tasks tobe performed by a plurality of independent entities comprising the stepsof a) generating a plurality of schedules for performance of theindependent tasks; b) submitting the plurality of schedules to a mastercoordinator for approval or disapproval of the schedules, resulting ingeneration of approval and/or disapproval decisions by the mastercoordinator; c) returning the approval and/or disapproval decisions tothe independent entities; d) adjusting the schedule for which adisapproval decision is returned, resulting in at least one adjustedschedule; and e) returning the adjusted schedule to the mastercoordinator for reconsideration. Steps b) through e) are repeated untilall of the schedules have been approved.

[0011] Benders decomposition with its mathematical feature is aparticularly suitable match for providing assistance to human operatorsfor supervisory control and coordination in the electric power industryin which there are a number of independent companies, generation,transmission, distribution, and customers, with their own objectives andthere is a coordinator, for example the ISO in the power grid, whichwill coordinate the proposed activities. The application of Bendersdecomposition enables coordination of various conflicting objectives indifferent power companies, thereby enabling maintenance of a certainlevel of reliability in the power system while individual companieswithin the system try to minimize their own cost of operation andmaximize their revenues.

[0012] Although disclosed herein as being applicable to deregulatedpower systems, it will be apparent to those skilled in the art that themethod of this invention employing Benders decomposition may be appliedto any system requiring supervisory control and coordination ofindependent tasks to be performed by independent entities, and suchapplications are deemed to be within the scope of this invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] These and other objects and features of this invention will bebetter understood from the following detailed description taken inconjunction with the drawings wherein:

[0014]FIG. 1 is diagram of an example of the Benders decompositionhierarchy in which the master problem represents the coordinator in thepower grid and sub-problems identify individual power companies;

[0015]FIG. 2 is a diagram of a Benders decomposition as applied tomaintenance scheduling;

[0016]FIG. 3 is a diagram showing application of the method of thisinvention as applied to multiple companies in a deregulated system; and

[0017]FIG. 4 is a diagram of load data for an exemplary three-bussystem.

DESCRIPTION OF PREFERRED EMBODIMENTS

[0018]FIG. 1 shows an example of the Benders decomposition hierarchy inwhich the master problem represents the coordinator in the power gridand sub-problems identify individual power companies. The individualcompanies schedule their own activities and submit their proposedschedules to the master problem which will check various criteria andapprove or disapprove the proposed schedules. The coordinator's decisionis returned as Benders cuts to sub-problems which will correspondinglyadjust their schedules if their proposals are not approved. Subsequentiterations follow before the implementation of proposed schedules untilall criteria are met.

[0019] In the past, when a power system was operated as a regulatedmonopoly, most techniques for thermal generator maintenance schedulingwere based on heuristic approaches. These approaches consider agenerating unit separately in selecting its optimal outage intervalsubject to constraints and an objective criterion for equalizing orleveling reserves throughout the planning interval, minimizing expectedtotal production costs or leveling the risk of failure to meet demand.An example of a heuristic approach would be to schedule one unit at atime beginning with the largest and ending with the smallest. Mostmethods, mainly those based on heuristics, represent only the generationsystem and do not take into account the network constraint effects onthe unit maintenance. The method of this invention extends Bendersdecomposition to include coordination between a GENCO and a TRANSCO forthe inclusion of network constraints in the maintenance schedulingproblem.

[0020] The maintenance scheduling problem will determine the period forwhich generating units of a GENCO should be taken off line for plantpreventive maintenance over the course of a one or two year planninghorizon so that the total operating cost is minimized while systemenergy and reliability requirements as well as the number of otherconstraints in TRANSCO are satisfied. Since GENCO and TRANSCO are twoseparate entities, TRANSCO's network constraints in maintenancescheduling of generating units may be excluded. The exclusion may resultin an optimistic solution which will not satisfy network flowconstraints and cannot be implemented since generating units in a GENCOare distributed throughout the network and interconnected bytransmission lines. This may lead to different risk levels to meetdemand for a given amount of maintenance capacity outage depending onthe unit location in the system. When network constraints are included,the problem becomes considerably more complex. The method of thisinvention is designed to address this problem.

[0021] In coordinating the activities of GENCO and TRANSCO through anISO, the objective is to minimize total operating and maintenance costsover the operating planning period, subject to unit maintenance andnetwork constraints. To calculate the maintenance schedule, it isessential that numerous and complex constraints which limit the choiceof scheduling times are incorporated into the solution method.Constraints in the maintenance scheduling problem are categorized ascoupling and decoupling constraints. A coupling constraint is therequirement that generating units be overhauled regularly. This isnecessary to maintain their efficiency at a reasonable level, maintainthe incidence of forced outages low, and prolong the life of units. Thisprocedure is incorporated periodically by specifying minimum/maximumtimes that a generating unit may run without maintenance. The timerequired for overhauling a unit is generally known, and hence the numberof weeks that a unit is “down” is predetermined. It is assumed forpurposes of this discussion that there is very little flexibility inmanpower usage for maintenance. Furthermore, only a limited number ofunits may be serviced at one time due to the limited manpower. Theavailable crew could be split into geographical and organizationaltypes. For purposes of this description, we assume that the number ofcrew in each type required at each stage of overhaul of each unit isspecified.

[0022] Network constraints in a TRANSCO in each time period areconsidered as decoupling constraints. The network can be modeled aseither the transportation model or a linearized power flow model. Forpurposes of this description, the transportation model is used torepresent TRANSCO's operating limits and the peak load balance equation.Mathematically, a complete generator maintenance schedule can beformulated as follows:${Min}{\sum\limits_{t}{\sum\limits_{i}\left\{ {{C_{it}\left( {1 - x_{it}} \right)} + {c_{it}g_{it}}} \right\}}}$S.T.

[0023] maintenance constraints:

x _(it)=1 for t≦e _(i) or t≧l _(i) +d _(i)

x _(it)=0 for s _(i) ≦t≦s _(i) +d _(i)

x _(it)=0 or 1 for e _(i) ≦t≦l _(i)  (i)

[0024] 1. crew avalibility 2. resources availability 3. seasonallimitations 4. desirable schedule system constraits: (ii)

Sƒ+g+r=d∀t(iii)

g≦g·x∀t(iv)

r≦d∀t(v)

{ƒ{≦ƒ∀t(vii)

[0025] $\begin{matrix}\begin{matrix}{{\sum\limits_{i}r_{it}} \leq ɛ} & {\forall t} & \quad\end{matrix} & ({vii})\end{matrix}$

[0026] where: C_(ii) generation maintenance cost for unit i at time tc_(it) generation cost of unit i at time t x_(it) unit maintenancestatus, 0 if unit is off-line for maintenance s_(i) period in whichmaintenance of generating unit i starts e_(i) earliest period formaintenance of generating unit i to begin l_(i) latest period formaintenance of generating unit i to begin d_(i) duration of maintenancefor generating unit i r vector of dummy generators which corresponds toenergy not served at time period t {overscore (f)} maximum line flowcapacity in matrix term f active power flow in vector term {overscore(g)} maximum generation capacity in vector term g vector of (g_(it))power generation for each unit at time t d vector of the demand in everybus at time t S node-branch incidence matrix ε acceptable level ofexpected energy not served

[0027] The unknown variable x_(it) in (2-1) is restricted to integervalues; on the other hand, g_(it) has continuous values. Therefore (2-1)corresponds to a mixed-integer programming problem.

[0028] The objective of (2-1) is to minimize the total maintenance andproduction cost in a GENCO over the operational planning period. Thefirst term of objective function (2-1) is the maintenance cost ofgenerators; the second is the energy production cost.

[0029] In (2-1), maintenance constraints (i) and (ii) represent GENCO'sconstraints. Also, constraints (i) represent the maintenance windowstated in terms of the start of maintenance variables (s_(i)). The unitmust not be in maintenance before its earliest period of maintenance(e_(i)) and latest period of maintenance (e.g., l_(i)+d_(i)). The set ofconstraints (ii) consists of crew and resources availability, seasonallimitations, desirable schedule, and other constraints such as fuel andenvironmental constraints. The seasonal limitations can be incorporatedin e_(i) and l_(i) values of constraint (i). If units 1, 2 and 3 are tobe maintained simultaneously, the set of constraints would be formed asfollows:

x _(1t) +x _(2t) +x _(3t)=3 or x _(1t) +x _(2t) +x _(3t)=0

[0030] If in each maintenance area we have limited resources and crewavailable, the set of constraints would be formed as follows:$\begin{matrix}{{\sum\limits_{i \in A}{\sigma_{m\quad i}\left( {1 - x_{it}} \right)}} \leq Z_{m\quad t}} & \text{(2-2)}\end{matrix}$

[0031] In the case of resource constraints, Z_(mt) would be the amountof resource m available in area A for each time t and σ_(mi) would be apercentage of this resource required for unit i. In the case of crewconstraints, the corresponding Z_(t) would be the number of maintenancecrew in area ω and σ_(mi) would be a percentage of this crew requiredfor maintenance of unit i.

[0032] Constraints (iii)-(vi) represent the operation constraintschecked by the ISO, in this case, peak load balance and other operationconstraints such as generation and transmission capacity limits of thesystem. Constraint (vii) represents allowable energy unserved in thesystem. Benders decomposition is there applied to solve GENCO's and ISOset of equations in (2-1).

Benders Decomposition

[0033] Before we discuss the solution methodology, let us present theBenders decomposition by considering the following general mixed integerprogram:

Min Px+p(g)

S.T. A ₁ x≦b ₁

A ₂ x+u(g)≦b ₂

g≧0

x _(i)=0 or 1 for all i  (3-1)

[0034] where x is a vector of 0-1 variables with constant cost vector Pand coefficient matrices A₁ and A₂; g is a vector of continuousvariables with cost functions p and u; b₁ and b₂ are vectors of righthand side constants. Since the problem involves both discrete andcontinuous variables, it is unlikely that a direct approach to solve(3-1) would be computationally feasible. Instead the problem ispartitioned as

Min Px+Min_(g≧0) {p(g)|u(g)≦b ₂ −A ₂ x}

S.T. A ₁ x≦b ₁

x _(i)=0 or 1

x∈Ω  (3-2)

[0035] where Ω is the set of x for which the constraints u(g)≦b₂-A₂x canbe satisfied. For each fixed x, the resulting inner minimization problemis

Min p(g)

S.T. u(g)≦b ₂ −A ₂ x

g≧0  (3-3)

[0036] The Lagrangian relaxation of (3-3) is given by

L(α)=min_(g≧0) {p(g)+α(u(g)−(b ₂ −A ₂ x))}  (3-4)

[0037] If g does satisfy u(g)≦b₂−A2x, the extra term in the objectivewill be non-positive and thus, for all α>0,

L(α)≦min_(g≧0) {p(g)|u(g)≦(b ₂ −A ₂ x)}  (3-5)

[0038] The Lagrangian dual L is then defined by L=max_(α)L(α). Undercertain conditions sufficient for strong duality,

L=min_(g≧0) {p(g)|u(g)≦(b ₂ −A ₂ x)}  (3-6)

[0039] enabling us to replace the inner minimization of (3-2) by L. Thisreplacement is justified later for our problem. With this replacement,(3-2) becomes

Min Px+Max_(α≧0) {L(α)}

S.T. A ₁ x≦b ₁

x _(i)=0 or 1

x∈Ω  (3-7)

[0040] If we let $\begin{matrix}{z = \quad {{Px} + {\max_{\alpha \geq 0}\left\{ {L(\alpha)} \right\}}}} \\{= \quad {{Px} + {\max_{\alpha \geq 0}\left\{ {\min_{g \geq 0}\left\{ {{p(g)} + {\alpha \left( {{u(g)} - \left( {b_{2} - {A_{2}x}} \right)} \right)}} \right\}} \right\}}}}\end{matrix}$

[0041] then (3-7) is equivalent to

Min z

S.T. A ₁ x≦b ₁

z≧Px+min_(g≧0) {p(g)+α(u(g)−(b ₂ −A ₂ x))} for all α

x _(i)=0 or 1

x∈Ω  (3-8)

[0042] Constraints (3-8) are referred to as feasibility cuts.

[0043] To complete the derivation of the master problem, the set ofconstraints that ensure x∈Ω must be characterized. This condition issatisfied if and only if

max_(β≧0){min_(g≧0) {p(g)+β(u(g)−(b ₂ −A ₂ x))}}≦∞

[0044] This condition is equivalent to

min_(g≧0){β(u(g)−(b ₂ −A ₂ x))}≦0 for all β≧0  (3-9)

[0045] Constraints of this form are referred to as infeasibility cuts.Thus our master problem is

Min z

S.T. z≧Px+min_(g≧0) {p(g)+α(u(g)−(b ₂ −A ₂ x))} for allα>0min_(g≧0){β(u(g)−(b ₂ −A ₂ x))}≦0 for all β≧0  (3-10)

[0046] x_(i)=0 or 1

[0047] The application of this method to our problem is discussed in thefollowing section.

Solution Methodology

[0048] Benders decomposition is applied to the generation maintenanceproblem as follows. If X denotes the vector of maintenance variables{x_(it)}, Ω represents the set of maintenance schedules for whichconstraints (iii)-(vii) are satisfied in all periods t, and operationcost w_(t) is defined as $w_{t} = {\sum\limits_{i}{c_{it}g_{it}}}$

[0049] then (2-1) can be written as $\begin{matrix}{\left. \left. {{{Min}{\sum\limits_{t}{\sum\limits_{i}{C_{it}\left( {1 - x_{it}} \right)}}}} + {\sum\limits_{t}{{Min}\left\{ {w_{t}{({iii}) - \left\{ {vii} \right.}} \right.}}} \right) \right\} {S.T.}} & \text{(4-1)}\end{matrix}$

[0050] maintenance constraints:

x _(it)=1 for t≦e _(i) or t≧l _(i) +d _(i)

x _(it)=0 for s_(i) ≦t≦s _(i) +d _(i)

x _(it)=0 or 1 for e _(i) ≦t≦l _(i)  (i)

[0051] 1. crew availability 2. resources availablity 3. seasonallimitations 4. desirable schedule(ii)

X∈Ω

[0052] If the t^(th) sub-problem was a linear program, it could bereplaced by its dual as is done in the standard Benders decomposition.The Lagrangian dual of the t^(th) sub-problem is given by

L _(t)=max_(κ,π,γ,ζ,μ≧0) {L _(t)(κ,π,γ,ζ,μ)}  (4-2)

[0053] where L_(t)(κ,π,γ,ζ,μ) is the Lagrangian function and κ,π,γ,ζ,μand μ are multipliers of constraints (iii)-(vii). $\begin{matrix}{{L_{t}\left( {\kappa,\pi,\gamma,\zeta,\mu} \right)} = {\min_{g \geq 0}\begin{Bmatrix}{w_{t} + {\sum\limits_{i}{\kappa_{it}\left( {{\sum\limits_{k}\left( {S_{ik}f_{kt}} \right)} + g_{it} + r_{it} - d_{it}} \right)}} +} \\{{\sum\limits_{i}{\pi_{it}\left( {g_{it} - {{\overset{\_}{g}}_{i} \cdot x_{it}}} \right)}} + {\sum\limits_{i}{\gamma_{it}\left( {r_{it} - d_{it}} \right)}} +} \\{{\sum\limits_{k}{\zeta_{kt}\left( {{f_{kt}} - {\overset{\_}{f}}_{k}} \right)}} + {\mu_{t}\left( {\left( {\sum\limits_{i}r_{it}} \right) - ɛ} \right)}}\end{Bmatrix}}} & \text{(4-3)}\end{matrix}$

[0054] The t^(th) sub-problem is then placed by L_(t), and (4-1) isrewritten as $\begin{matrix}{{{{Min}{\sum\limits_{t}{\sum\limits_{i}{C_{it}\left( {1 - x_{it}} \right)}}}} + L_{t}}{S.T.}} & \text{(4-4)}\end{matrix}$

[0055] maintenance constraints:

x _(it)=1 for t≦e _(i) or t≧l _(i) +d _(i)

x _(it)=0 for s _(i) ≦t≦s _(i) +d _(i)

x _(it)=0 or 1 for e _(i) ≦t≦l _(i)  (i) 1. crew availability 3.seasonal limitations 2. resources availability 4. preschedule (ii)

X∈Ω

[0056] To ensure X∈Ω, the maintenance schedule must ensure thatsufficient reserve exists to provide a secure supply while minimizingthe cost of operation. The t^(th) sub-problem is feasible if and only ifthe optimal value of the following problem is less than ∈$\begin{matrix}{{{Min}{\sum\limits_{i}r_{it}}}\begin{matrix}{{{S.T.{\quad \quad}{Sf}} + g + r} = d} \\{\quad {g \leq {\overset{\_}{g} \cdot x}}} \\{\quad {r \leq d}} \\{\quad {{f} \leq \overset{\_}{f}}}\end{matrix}} & \text{(4-5)}\end{matrix}$

[0057] Its Lagrangian dual is

max_(ν,λ,τ,η≧0) U _(t)(ν,λ,τ,η)

[0058] where U_(t)(ν,λ,τ,η) is the following Lagrangian function andν,λ,τ and η are multipliers of constraints (iii)-(vii).${U_{t}\left( {v,\lambda,\tau,\eta} \right)} = {\min_{g \geq 0}\begin{Bmatrix}{{\sum\limits_{i}r_{it}} + {\sum\limits_{i}{v_{it}\left( {\left( {\sum\limits_{k}{S_{ik}f_{kt}}} \right) + g_{it} + r_{it} - d_{it}} \right)}} +} \\{{\sum\limits_{i}{\lambda_{it}\left( {g_{it} - {{\overset{\_}{g}}_{i} \cdot x_{it}}} \right)}} + {\sum\limits_{i}{\tau_{it}\left( {r_{it} - d_{it}} \right)}} +} \\{\sum\limits_{k}{\eta_{kt}\left( {{f_{kt}} - {\overset{\_}{f}}_{k}} \right)}}\end{Bmatrix}}$

[0059] We then arrive at the generalized Benders master problem:Min  z${{S.T.z} \geq {{\sum\limits_{t}{\sum\limits_{i}{C_{it}\left( {1 - x_{it}} \right)}}} + {\sum\limits_{t}{L_{t}\left( {\kappa,\pi,\gamma,\zeta,\mu} \right)\quad {for}\quad {all}\quad \kappa}}}},\pi,\gamma,\zeta,{\mu \geq 0}$${{\sum\limits_{t}{U_{t}\left( {v,\lambda,\tau,\eta} \right)}} \leq {ɛ\quad {for}\quad {all}\quad v}},\lambda,\tau,{\eta \geq 0}$

[0060] maintenance constraints:

x _(it)=1 for t≦e _(i) or t≧l _(i) +d _(i)

x _(it)=0 for s _(i) ≦t≦s _(i) +d _(i)

x _(it)=0 or 1 for e _(i) ≦t≦l _(i)  (i)(4-6) 1. crew availability 3.seasonal limitations 2. resources availability 4. desirable schedule(ii)

[0061] The problem is decomposed into a master problem and a set ofindependent operation sub-problems. The master problem, which in thismodel is an integer programming problem, is solved to generate a trialsolution for maintenance schedule decision variables. This masterproblem is a relaxation of the original problem in that it contains onlya subset of constraints. Its optimal objective value is a lower bound onthe optimal value of the original problem. Once x_(it) variables arefixed, the resulting operation sub-problem can be treated as a set ofindependent sub-problems, one for each time period t, since there are noconstraints across time periods. The set of operation sub-problems arethen solved using the fixed maintenance schedule obtained from thesolution of the master problem. At each iteration, the solution ofsub-problems generates dual multipliers which measure the change ineither production cost or reliability resulting from marginal changes inthe trial maintenance scheduling. These dual multipliers are used toform one or more constraints (known as cuts) which are added to themaster problem for the next iteration. The problem continues until afeasible solution is found whose cost is sufficiently close to lowerbound, as shown in FIG. 2.

[0062] The initial maintenance master problem is formulated as follows:Min  z${S.T.\quad z} \geq {\sum\limits_{t}{\sum\limits_{i}\left\{ {C_{it}\left( {1 - x_{it}} \right)} \right\}}}$

[0063] maintenance constraints:

x _(it)=1 for t≦e _(i) or t≧l _(i) +d _(i)

x _(it)=0 for s _(i) ≦t≦s _(i) +d _(i)

x _(it)=0 or 1 for e _(i) ≦t≦l _(i)  (4-7)

[0064] 1. crew availability 3. seasonal limitations 2. resourcesavailability 4. desirable schedule

Operation Sub-Problem

[0065] The sub-problem may not have any solutions due to the fact thatthe unserved energy cannot be kept above a desired level. If asub-problem is infeasible, then an infeasibility cut is generated. Foreach infeasible sub-problem resulting from the nth trial solution of themaster problem, the infeasible cut is of the form $\begin{matrix}{{{\sum\limits_{i}r_{it}^{n}} + {\sum\limits_{i}{\lambda_{it}^{n}{{\overset{\_}{g}}_{i}\left( {x_{it}^{n} - x_{it}} \right)}}}} \leq ɛ} & \text{(4-8)}\end{matrix}$

[0066] The multiplier λ_(it) ^(n) may be interpreted as a marginaldecrease in energy not supplied with a 1 MW increase in generation,given the nth trial maintenance schedule. The infeasibility cuts (4-8)will eliminate maintenance values which are not possible to bescheduled.

[0067] If the sub-problem is feasible, then the fuel cost for period t,w_(t), depends on the utilization of the available units to satisfy loadconstraints in each time period subject to maintaining the unservedenergy above a certain level. Thus the generation cost in period t canbe expressed as $\begin{matrix}{{w_{t} = {{Min}{\sum\limits_{i}{c_{it}g_{it}}}}}{{{S.T.\quad {Sf}} + g + r} = d}\quad {g \leq {\overset{\_}{g}x^{n}}}\quad {r \leq d}\quad {{f} \leq \overset{\_}{f}}\quad {{\sum\limits_{i}r_{it}} \leq ɛ}} & \text{(4-9)}\end{matrix}$

[0068] The solution of the sub-problem is not complicated, since knowingwhich generators and transmissions are available during period t allowsus to minimize the operation cost. The feasible cut is of the form$\begin{matrix}{z \geq {\sum\limits_{t}\left( {w_{t}^{n} + {\sum\limits_{i}\left( {{C_{it}\left( {1 - x_{it}^{n}} \right)} + {\pi_{it}^{n}{{\overset{\_}{g}}_{i}\left( {x_{it}^{n} - x_{it}} \right)}}} \right)}} \right)}} & \text{(4-10)}\end{matrix}$

[0069] where w_(t) ^(n) is the expected fuel cost for period tassociated with the n^(th) trial solution. The multiplier π_(it) ^(n)may be interpreted as the marginal cost associated with a 1 MW decreasein the power capacity, given the n^(th) trial maintenance schedule. Thecost cuts (4-10) will tend to increase the lower bounds obtained fromeach successive maintenance sub-problem solution.

Maintenance Master Problem

[0070] The maintenance master problem minimizes maintenance cost subjectto maintenance constraints as well as feasibility and infeasibility cutsfrom the operation sub-problems. If all sub-problems are feasible, thentheir solutions yield a set of dual multipliers from which a feasibilitycut is constructed. If one or more operation sub-problems areinfeasible, then, for each infeasible sub-problem, an infeasibility cutis generated. Min  z${S.T.z} \geq {\sum\limits_{t}{\sum\limits_{i}\left\{ {C_{it}\left( {1 - x_{it}} \right)} \right\}}}$

[0071] maintenance constraints:

x _(it)=1 for t≦e _(i) or t≧l _(i) +d _(i)

x _(it)=0 for s _(i) ≦t≦s _(i) +d _(i)

x _(it)=0 or 1 for e _(i) ≦t≦l _(i)

[0072] 1. crew availability 2. resources availability 3. seasonallimitations 4. desirable schedule

[0073] feasibility and infeasibility cuts from previous iterations

[0074] if all sub-problems are feasible then the feasible cut is:$z \geq {\sum\limits_{t}\left( {w_{r}^{n} + {\sum\limits_{i}{C_{it}\left( {1 - x_{it}^{n}} \right)}} + {\pi_{it}^{n}{{\overset{\_}{g}}_{i}\left( {x_{it}^{n} - x_{it}} \right)}}} \right)}$

[0075] if one or more sub-problems are infeasible, then the infeasiblecuts are: $\begin{matrix}{{{{\sum\limits_{i}r_{it}^{n}} + {\sum\limits_{i}{\lambda_{it}^{n}{{\overset{\_}{g}}_{i}\left( {x_{it}^{n} - x_{it}} \right)}}}} \leq ɛ}\quad {\forall{t \in {\text{infeasible}\quad \text{sub-problem}}}}{{{x\varepsilon}\quad 0},1}} & \text{(4-11)}\end{matrix}$

[0076] where

[0077] n is the current number of iterations

[0078] λ^(n), π^(n) are the multiplier vectors at n^(th) iteration

[0079] The important feature of the Benders decomposition is theavailability of upper and lower bounds to the optimal solution at eachiteration. These bounds can be used as an effective convergencecriterion. The critical point in the decomposition is the modificationof objective function based on the solution of the operationsub-problem. Associated with the solution of the operation sub-problemis a set of dual multipliers which measure changes in system operatingcosts caused by marginal changes in the trial maintenance. Thesemultipliers are used to form a linear constraint, written in terms ofmaintenance variable x. This constraint, known as Benders cut, isreturned to the maintenance problem which is modified and solved againto determine a new trial maintenance plan.

EXAMPLE

[0080] We use a three-bus system as an example. The maximum energy notserved requirement (∈) is 0 p.u., and generator and line input data inper unit for GENCO and TRANSCO are given in Tables 1 and 2. Load dataare depicted in FIG. 2. , GENCO will perform maintenance on at least onegenerator. We assume the study period represents one time interval.Loads are assumed constant during the study period. TABLE 1 GeneratorData (GENCO) Min Capacity Max Capacity Maint. Cost/Unit Unit (p.u.)(p.u.) Cost ($) ($) 1 0.5 2.5 10 g₁ 300 2 0.6 2.5 10 g₂ 200 3 0.6 3.0 10g₃ 100

[0081] TABLE 2 Line Data (TRANSCO) Line Ω/line # of lines Capacity/line(p.u.) 1-2 0.2 2 0.25 2-3 0.25 2 0.5 1-3 0.4 2 0.25

[0082] First, we solve the initial maintenance master problem.

[0083] GENCO's Maintenance Problem (iteration 1):

Min z

S.T. 300*(1−x ₁)+200*(1−x ₂)+100*(1−x ₃)≦z

x ₁ +x ₂ +x ₃≦2

X ₁≦1 x ₂≦1 x ₃≦1

[0084] The solution is:

[0085] x₁=1

[0086] x₂=1

[0087] x₃=0

[0088] z=100

[0089] ISO's operation problem (iteration 1):

[0090] We check the feasibility of the operation sub-problem given thefirst trial of maintenance schedule. The feasibility check is given asfollows: Min r₁ + r₂ + r₃ S.T. −f₁₂ − f₁₃ + g₁ + r₁ = 1 Load balance atbus 1 −f₂₃ + f₁₂ + g₂ + r₂ = 3 Load balance at bus 2 f₁₃ + f₂₃ + g₃ + r₃= 1 Load balance at bus 3 0.5 ≦ g₁ ≦ 2.5 Generator 1 limit 0.6 ≦ g₂ ≦2.5 Generator 2 limit 0.0 ≦ g₃ ≦ 0.0 Generator 3 limit −2*0.25 ≦ f₁₂ ≦2*0.25 Line 1-2 flow limit −2*0.25 ≦ f₁₃ ≦ 2*0.25 Line 1-3 flow limit−2*0.5 ≦ f₂₃ ≦ 2*0.5 Line 2-3 flow limit

[0091] The primal solution of the feasibility check is:

[0092] r=0.5

[0093] g₁=2

[0094] g₂=2.5

[0095] g₃=0

[0096] f₁₂=0.5

[0097] f₁₃=0.5

[0098] f₂₃=0

[0099] The dual price of the operation sub-problem is:

[0100] λ_(g1)=0

[0101] λ_(g2)=1

[0102] λ_(g3)=1

[0103] The above LP solution is infeasible, since r₁+r₂+r₃≧0. Thegeneration cost is set arbitrarily to 1000 because the solution isinfeasible. The infeasible cut is:

0.5+1*2.5(1−x ₂)+1*3*(0−x ₃)≦0

[0104] GENCO's Maintenance Problem (Iteration 2):

Min z

S.T. 300*(1−x ₁)+200*(1−x ₂)+100*(1−x ₃)≦z 0.5+1*2.5*(1−x ₂)+1*3*(0−x₃)≦0.5 x ₁ +x ₂ +x ₃≦2 x ₁≦1 x ₂≦1 x ₃≦1

[0105] The solution is:

[0106] x₁=1

[0107] x₂=0

[0108] x₃=1

[0109] z=200

[0110] ISO's Operation Problem (Iteration 2):

[0111] The feasibility check is as follows: Min r₁ + r₂ + r₃ S.T. −f₁₂ −f₁₃ + g₁ + r₁ = 1 Load balance at bus 1 −f₂₃ + f₁₂ + g₂ + r₂ = 3 Loadbalance at bus 2 f₁₃ + f₂₃ + g₃ + r₃ = 1 Load balance at bus 3 0.5 ≦ g₁≦ 2.5 Generator 1 limit 0.0 ≦ g₂ ≦ 0.0 Generator 2 limit 0.6 ≦ g₃ ≦ 3.0Generator 3 limit −2*0.25 ≦ f₁₂ ≦ 2*0.25 Line 1-2 flow limit −2*0.25 ≦f₁₃ ≦ 2*0.25 Line 1-3 flow limit −2*0.5 ≦ f₂₃ ≦ 2*0.5 Line 2-3 flowlimit

[0112] The primal solution of feasibility check is:

[0113] r=1.5

[0114] g₁=1.5

[0115] g₂=0

[0116] g₃=2

[0117] f₁₂=0.5

[0118] f₁₃=0

[0119] f₂₃=−1

[0120] The dual price of the operation sub-problem is:

[0121] λ_(g1)=0

[0122] λ_(g2)=1

[0123] λ_(g3)=0

[0124] The above LP solution is infeasible, since r₁+r₂+r₃≧0. Thegeneration cost is set arbitrarily to 1000 because the solution isinfeasible. The infeasible cut is as follows:

1.5+1*2.5*(1−x ₂)≦0

[0125] GENCO's Maintenance Problem (Iteration 3):

Min z

S.T. 300*(1−x ₁)+200*(1−xg ₂)+100*(1−xg ₃)≦z

0.5+1*2.5*(1−x ₂)+1*3*(0−x ₃)≦0.5

1.5+1*2.5*(1−x ₂)≦0

x ₁ +x ₂ +x ₃≦2

x₁≦1 x ₂≦1 x ₃≦1

[0126] The solution is:

[0127] x₁=0

[0128] x₂=1

[0129] x₃=1

[0130] z=300

[0131] Given the trial maintenance schedule in iteration 3, we apply thefeasibility check as before which gives t-0. This means the trialschedule is feasible now.

[0132] ISO's Operation Problem (Iteration 3):

[0133] The feasible problem is as follows Min w = 10*g₂ + 10*g₃ + 300S.T. −f₁₂ − f₁₃ + g₁ + r₁ = 1 Load balance at bus 1 −f₂₃ + f₁₂ + g₂ + r₂= 3 Load balance at bus 2 f₁₃ + f₂₃ + g₃ + r₃ = 1 Load balance at bus 30.0 ≦ g₁ ≦ 0.0 Generator 1 limit 0.6 ≦ g₂ ≦ 2.5 Generator 2 limit 0.6 ≦g₃ ≦ 3.0 Generator 3 limit −2*0.25 ≦ f₁₂ ≦ 2*0.25 Line 1-2 flow limit−2*0.25 ≦ f₁₃ ≦ 2*0.25 Line 1-3 flow limit −2*0.5 ≦ f₂₃ ≦ 2*0.5 Line 2-3flow limit r₁ + r₂ + r₃ ≦ 0

[0134] The primal solution is:

[0135] w=350

[0136] g₂=2.5

[0137] g₃=2.5

[0138] f₁₂=−0.5

[0139] f₁₃=−0.5

[0140] f₂₃=−1

[0141] The dual price of the operation sub-problem is:

[0142] π_(g1)=0

[0143] π_(g2)=−10

[0144] π_(g3)=−10

[0145] The feasible cut for the third iteration is:

[0146] z≧350−10*2.5*(1−x ₂)−10*2.5*(1−x ₃)

[0147] GENCO's Maintenance Problem (Iteration 4):

Min z

S.T. 300*(1−x ₁)+200*(1−x ₂)+100*(1−x ₃)≦z

0.5+1*2.5*(1−x ₂)+1*3*(0−x ₃)≦0.5

1.5+1*2.5*(1−x ₂)≦0

z≧350−10*2.5*(1−x ₂)−10*2.5*(1−x ₃)

x ₁ +x ₂ +x ₃≦2

x ₁≦1 x ₂≦1 x ₃≦1

[0148] The solution is:

[0149] x₁=0

[0150] x₂=1

[0151] x₃=1

[0152] z=350

[0153] We stop here since z=w which means the cost is equal to the lowerbound.

[0154] While in the foregoing specification this invention has beendescribed in relation to certain preferred embodiments thereof, and manydetails have been set forth for purpose of illustration, it will beapparent to those skilled in the art that the invention is susceptibleto additional embodiments and that certain of the details describedherein can be varied considerably without departing from the basicprinciples of the invention.

I claim:
 1. A method for supervisory control and coordination ofindependent tasks to be performed by a plurality of independent entitiescomprising the steps of: a) generating a plurality of schedules forperformance of said independent tasks; b) submitting said plurality ofschedules to a master coordinator for one of approval and disapproval ofsaid schedules, resulting in generation of a plurality of at least oneof an approval decision and a disapproval decision by said mastercoordinator; c) returning said plurality of said at least one of anapproval decision and a disapproval decision to said independententities; d) adjusting said schedule for which a disapproval decision isreturned, resulting in at least one adjusted schedule; and e) returningsaid at least one adjusted schedule to said master coordinator forreconsideration.
 2. A method in accordance with claim 1, wherein stepsb) through e) are repeated until all said schedules have been approved.3. A method in accordance with claim 2, wherein said schedules areimplemented after all said schedules have been approved.
 4. A method inaccordance with claim 1, wherein said master coordinator is a powercoordinator in a power grid.
 5. A method in accordance with claim 4,wherein said independent entities are individual power companies.